An Abstract Condensation Property

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An Abs

trac

t Condensation Property

Thesis by David Law

In

Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

California Institute of Technology Pasadena, California

1994

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@1994

David Law

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ACKNOWLEDGMENTS

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ABSTRACT

Let A

= (A, ...

) be a relational structure. Say that A has condensation if there is an F : A <w --+ A such that for every partial order P, it is forced by P that substructures of P which are closed under F are isomorphic to elements of the ground model. Condensation holds if every structure in V, the universe of sets, has condensation. This property, isolated by Woodin, captures part of

the content of the condensation lemmas for L, K and other "L-like" models. We present a variety of results having to do with condensation in this abstract sense. Section 1 establishes the absoluteness of condensation and some of its consequences. In particular, we show that if condensation holds in M, then M

F

GC H and there are no measurable cardinals or precipitous ideals in M. The results of this section are due to Woodin. Section 2 contains a proof that condensation implies OK (E) for /'i, regular and E ~ /'i, stationary. This is the main result of this thesis. The argument provides a new proof of the key lemma giving GC H. Section 2 also contains some information about the relationship between condensation and strengthenings of diamond. Section 3 contains partial results having to do with forcing "Cand(A)", some further discussion of the relation between condensation and combinatorial principles

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IAI<w

~

IAI.

Cond(Aj F) is the statement: For every partial order (p.o.) IP

IPII-VX.X

--<F

A

~:38 E

V.A

r

X ~

B

In other words, for any p.o. IP it is forced by IP that F-closed substructures

of A are isomorphic to structures in V. Cond(A) holds iff 3F.Cond(Aj F).

We will be concerned with condensation for structures of the form

A

=

(X, E, ... ) where X is a transitive set. (Notations for such structures will

typically suppress the

"E".)

C ond( Aj F) will hold for such structures just in

case F-closed substructures collapse to elements of V.

If Cond(A) holds we say that A has condensation. Let M be a transitive

model of ZFC. If M

F=

Cond(A), A has condensation in M. Condensation

holds in M iff M

F=

Va Cond(Vo). Equivalently, since M

F=

AC,

condensa-tion holds in M iff M

F=

VKVA ~ K Cond(K, A). For ordinals a, Cond(a) iff

VA ~ a. Cond(a, A). And if A is a subset of a, Cond(A) iff Cond(a, A) (iff

Cond( (a, E, A))).

Any other uses of the term "condensation" should be self-explanatory.

Section 1 establishes the absoluteness of condensation and some of its

consequences. In particular, we show that if condensation holds in M, then

M

F=

eCH and there are no measurable cardinals or precipitous ideals in M. The results of this section are due to Woodin [Wo]. Section 2 contains a proof

that condensation implies O,,(E) for K regular and E ~ K stationary. This is the main result of this thesis. The argument provides a new proof of the

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relationship between condensation and strengthenings of diamond. Section

3 contains partial results having to do with forcing "Cond(A)", some further

discussion of the relation between condensation and combinatorial principles

which hold in L, and an argument that Cond( G) fails in V[G], where G is

generic for the partial order adding W2 cohen subsets of WI •

Notations are fairly standard. [X]" is the collection of subsets of X of size

K..

[X]<"

is the collection of subsets of

X

of size less than K..

X<w

is the

collection of finite sequences from X. H" is the collection of sets hereditarily

of size less than K..

HC

=

Hw,.

If F

:

X<w

---+

X, C F

=

{U

C;

X

I

F"U<w

C;

U}.

(Occasionally,

CF

will refer to closed sets meeting some size

restriction.)

clF(U)

is the closure of

U

under

F

.

For any set

X,

trx

is the

collapsing map associated with

X.

C;e refers to end-extension. Coll( K:,

X)

is the partial order adding a function

f :

K. ~ X with conditions of size

<

K.. If t is a set-theoretic expression (t)M is the result of evaluating tin

M.

Similarly, if <I> is a formula (<I»M is the restriction of <I> to M. Thus "(<I»M"

and

14M

F

cP"

have the same meaning. In expository contexts

'

V'

refers to

the universe of sets. In statements of a forcing language 'V' refers to the

ground model. Other notations will be handled as they arise.

We make a few preliminary remarks and observations. If

F,

G :

X<w

---+

X define F :::; G iff G ~ F iff C_cC;

CF.

Lemma 0.1. If Cond(A; F) and F :::; G then Cond(A; G). ..,

Say that

F

is efficient if

clF(U)

= F"U<w.

It is sometimes convenient to

know that

Lemma 0.2. For any F there i.! an efficient G ~ F.

Proof. Define

cPk :

V<w ---+ V by

cPk(

8)

=

8k

if k

<

e( 8), and

cPk(

s)

=

80

if

k

~ e(8). Let;:k be the closure under composition of

{F, cPo,

...

cPd

and

:F

=

Uk

;:k.

Let (Fn

I

n

<

w) list;: so that if p(n) is the least k such that

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Condensation for sets of size S; WI is trivial. Thus the structures

A

which

come up for attention will have domains at least that size when viewed

from within a model in which Cond(A) is to be evaluated. Another property

common to the domains of structures we consider is closure under u f-> rk( u).

So in every case we will have rk(A)

=

A

n

OR ~ w. Also one may always

assume of a potential witness F to Cond(A) that clF(U)

n

Q+

=

Q where

Q = min(OR \ clF(U».

Lemma 0.3. The following are equivalent:

(1) Cond(A).

(2) Let Q = rk(A). There is a u such that for every () ~ Q and p.o. IP',

letting N = VB, IP' II-

'Ix.

u,A E X

-<

N

==>

7rx(A) E N.

(3) There is a u and a ()

>

Q such that for every p. o. IP', letting N

=

VB, IP' II-

'Ix.

u,A E X

-<

N

==>

7rx(A) E N.

Proof. Let A = (A, E, R, ... ).

(1)

==>

(2).

Let u = F.

(2)

==>

(3). Trivial.

(3)

==>

(1). Pick u and let F be an efficient Skolem function for (VB, {A, u } ). Let G: A<w --+ A be defined by G(s) = F(s) if F(s) E A and G(s) = 0

otherwise. If X E CG then AnclF(X)

=

X. So 7rx(A)

=

7rclF(x)(A), where

7rx(A)

=

(7rX II A, 7rX" E nA2, 7r x" R, ... ).

Thus C ond( Ai G) holds. -1

Lemma 0.4. If A is a transitive set, Cond(A) holds and B E A, then Cond(B).

Proof. Take u as in 0.3(2) and let ii = {u,A}. If ii,B E X

-<

VB then

u, A E .Y. And 7rx(B) E {7rx(v) I v E A

n

X}

=

7rx(A). So Cond(B) holds

by 0.3. -1

Lemma 0.5. The following are equivalent:

(1) For all Q, Cond(V_{o ).}

(2)

{Q

I

Cond(Vo )} is unbounded in OR.

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Proof. Clear.

-1

It should be noted in connection with Lemma 0.5 that, conceivably,

con-densation may hold for some set yet fail for every set of ordinals which code

that set. We touch upon this point again briefly in section l.

Finally,

Lemma 0.6. Let A be a 3et of ordinal3 and B E L[A]. Then Cond(A) imp/ie" Cond(B).

Proof. Choose

B

>

sup (a) so that B E Lo[A]. If A, B E X

-<

Vo then

7l"x(B) E L[7l"x(A)]. Apply lemma 0.3(3) with this Band u

=

{F, A}, with F

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Section 1

§1.1 Absoluteness

Lemma 1.1. Let M be a tran3itive model of ZFC, F : ",,<w --> "", A <:::; ""

with F, A E M. Supp03e that there i3 an X E CF 3uch that (X, X

n

A) collap3e3 to (A, A) fJ: M. Then there i3 3uch an X in M[G] where G i3

M-generic for P

=

Coll(w, p(",,)M).

Proof. Work in a model in which p(p)nM is countable. Let G be P-generic over M. And let (*) be the statement: 3X.X E CF and 1l'x"A fJ: p(",,)M. In M[G], let

x

be a real coding (A, F, p(",,)M) in such a way that it is ~l(x) to decide whether X E CF and rrx"A E p(",,)M. Then, (*) holds and is

L:l(x)

,

hence holds in M[G] by absoluteness of L:l relations. Thus

(P

II- (*

»M.

-1

Corollary 1.2. If Cond(A) fail3 in M , then for any F E M there i3 a counterexample to Cond(A; F) in M[G], where G i3 Coll(w, P(",,)M)-generic

over M.

Proof. Assume Cond(A), A <:::; "" fails in M. Let F : ",,<w --> "" E M. Pick

Q E

M such that

(Q

II- 3X.X E CF & 1l'X" A fJ: V)M. Let H be Q-generic

over M and apply lemma 1.1 in M[H].

-1

Remark: In Lemma 1.1 it is not necessary that M satisfy all of ZFC since the facts about forcing, and L:l-absoluteness used in the proof hold in a sufficiently strong finite fragment of Z FC.

It may be worth drawing the analogy with condensation in L at this point.

L:1-substructures of limit levels of L, wherever they are found, collapse to

elements of L. Similarly if M

F

Cond(A; F) then an F-closed substructure

of A in any end-extension of M collapses to an element of M. Cond(A; F) asserts that this is true for generic extensions. With this restriction the

preceding property of L is first-order expressible. Lemma 1.1 shows that this restriction is only apparent for suitable

M.

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Lemma 1.3. Cond(Aj F) iff L[A, F]

F=

Cond(Aj F).

Proof. The "if' direction follows from lemma 1.1: if there is a

counterexam-ple to Cond(Aj F) which is V-generic, then there is one which is L[A,

F]-generic.

So suppose L[A, F]

F=

-,Cond(Aj F). Let M

=

L[A, F] and K. be such that

A C; K., F : K.<w ----t K.. Since M

F=

..,Cond(Aj F), there are P E M and

P-names a,T E M, such that in M, P If- (T E CF & 17 = 7r/'A & 17 f/:. V). If

G,

H are filters on P which are mutually generic over M, then aG

-I-

17 H . Let

Q

be a forcing notion which makes both p(p)M and (true) P(K.) countable. If G is Q-generic over V then in V[G], there is a collection C of size 2W

consisting of filters on P which are pairwise mutually generic over M. The

map H I-> aH is I-Ion C. Since IP(K.)VI = w, there is a filter H E C such

that aH f/:. V. But TH E CF. Thus it is not forced by

Q

that images of A by collapses of F-closed sets lie in V. So Cond(Aj F) fails . ..,

The next result improves on corollary 1.2.

Lemma 1.4. Let A, F E M C; N, where M and N are models of (a strong

enough fragment of) ZFC. Assume that M

F=

-'Cond(Aj F) and that there

is a real in N \ M. Then in N there is a counterexample to Cond(Aj F)M

-i. e., an X E

C

F with 7r

x"

A f/:. M. Furthermore, X can be found with

M -countable order-type.

Proof. Work in M. Let B

>

K. be such that VB reflects "Cond(Aj

F)"

together

with enough of ZFC. Let P

=

Coll(w, P(K.» and find W with K., A, F, P E

W -( VB, IWI

=

w. Let 7r

=

7rw : W ----t M be the transitive collapse of W

and 7r(K., A, F, P)

=

(K, A, F, P). So M

F=

-,Cond(Aj F) and by Lemma 1.1,

M

F=

"P If- 3X E _{CF .}7r x" A f/:.

v."

In M there are P-names 17, T such that

Plf-(TC;K & TEC_F & a=7r/'A & af/:.V). Thenamesa,Tcan

be used to construct a "continuous" 1-1 map f: W2 ----t {(TG,aG)

I

G is

P-generic over

M}

so that given (f(u»J

=

aGu' U can be recovered. Then

(f(u»o = _{TG u lifts}to an F-closed set via 7r, 7r-J"TGu' which collapses to

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Define a map S f-+ (p.,a.) from 2<w into P x OR so that

(1) s.lt = : } P • .lPIl

(2) s ~ t = : } PI ~ p.,

(3) Pib II- a.

ti

17 and P'-l II- a. E 17 and,

(4)

for s

E

w2, (P.

I

s ~ x) is M-generic (generates a P-generic filter over

M).

Let (Dn

I

nEw) list the dense subsets of P in M. Given P. choose a.

and P.-i. i = 0,1 as follows. Since P II- 17

ti

V there is an ordinal 0 and there are conditions q, r ~ p. such that q II- a

ti

17 and r II- a E 17. (Since p. does not decide " 0 E 17", a

i=

au for u S;; s.) Now pick

if

~ q and

r

~ r in Dl (.),

where f(s)

=

length of s, and let a.

= a, p.

-o

=

if,

and P'-l

=

r.

If u E w2, let Gu

=

{p

E

Pip

is compatible with P. for some s ~ u}

and let feu)

=

(TG.,I7G.). Gu is M-generic. Since (P II- T E CF)M, TG.

is F-closed and 7r-IIITG. is F-closed with collapse I7G •. u can be recovered

from I7G. recursively. Thus if u E W2

n

N \ M then X = TG. is an F-closed set in N wi th 7r X II A.

=

17 G.

ti

M. And the order- type of X is countable in

M. .,

Now as an improvement of corollary l.2 we have,

Corollary 1.5. If Cond(A) fail3 in M, then for any F E M there is a

counterexample to Cond(A; F) in M[G] which i3 of countable order-type in

M where G i3 (2<w,~)-generic over M . .,

§1.2 Some consequences of condensation.

Lemma 1.6. If a i3 countable, A ~ pea) and Cond(A), then

IAI

~ WI.

We present two arguments for this lemma in this section and a third

ar-gument in section 2. Each argument is of interest. The second introduces a

fact used again in section 3.

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Theorem 1.7. Let K be an infinite cardinal. Cond(P(K» implie3 2"

=

K+. Hence conden3ation implie3 GC H.

Proof. Let P

=

Coll(w, K) and G be P-generic. Apply lemma 1.6 in V[G] taking A

=

P(K)V. If V

1=

IP(K)I

~ K++ then V[G]

1=

IAI ~ W2' By lemma

1.6, Cond(A) must fail in V[G], hence in V by absoluteness. -i

Corollary 1.8. CH i3 equivalent to Cond(P(w». -i

Proof of 1.6. Choose F: A<w ---+ A to witness Cond(A) so that if X E CF

then X is transitive. This is possible since a is countable. Suppose A ~ W2.

Let (A'I

I 17

<

8) list A without repetition, 8 ~ W2' Let G be generic for Namba forcing

([J],

p.289). So V[G]

1=

IWI vI

=

WI

and Cf(W2 V)

=

w. In V[G], let 5 <;;: W2 V be cofinal with order-type w. And let X

=

elF( {A'I

117

E

S}). Since Cond(A; F) holds and X is transitive, X E V. In V,

IXI

~ W2

since

{17

I

A'I E

X}

is cofinal in W2' Thus V[G]

1=

IXI

~ IW2VI

=

WI'

But in V[G], X is the closure of a countable set, hence countable. Contradiction.

-i

The next argument depends on the following fact.

Lemma 1.9. Let M, N be tran3itive model3 of ZFC, M <;;: N, F: A<W -+

A, FE M, M

1=

IAI ~ W2. If there i3 a real in N\M then CFnN\M

=I

0 .

Proof. (This is adapted from [V] where a stronger statement is proved.) Let h : 8 ---+ A be a bijection. Extend h to 8<w by h(

5)

= (h(

50),

.

..

h( 5n» and let

F

=

h-I

0 F 0 h : 8<w ---+ 8. Let G be the following two person game of length w : player I plays intervals In

=

[an,,Bn] <;;: W2 and player II plays ordinals "in

<

W2 so that ,Bn

<

"in

<

an+l. Player I moves first and wins a play iff el F( an

I

n

<

w)

n

W2 <;;: UnIn. This game is open for player II, hence determined, by Gale-Stewart.

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ao

=

O. And let In

=

(1(10, . • • , In). Since 1_{0 ,}••• , In, (1 E X n

+

₁

-<

VB, it

follows that In

<

an+l, Thus ((1n, In)

In

<

w) is a run of G which player I

wins, since clf(a_n

In <

w) ~ Unln. So (1 is not winning for II.

So player I has a winning strategy (1 • Let 10

<

II

< ... <

W2 be the

first w ordinals closed under each of a ,F, and h (hence under

F)

in the sense

that, e.g., clFC'Yi)

n

W2 ~ Ii.

If

a ~ w, let

Xa

= clf(a

n

I

n

<

w) where

(an'

!3nl

=

In is obtained by using (1 against II's play (,n

I

n E a). Let

Xa

=

h"X

a. Then n E a iff Xa nC'Yn" n+l)

#

0. So a can be recovered

from Xa.

Carrying out this argument in M and using a real a E N \ M yields a set

Xa E CF

n

N \ M, hence lemma 1.9. .,

Proof of 1.6 (2nd). Choose F as before. The result is immediate from lemma

1.9 . .,

Theorem 1. 7 is not stated optimally. It is easy to see that C ond(P( K))

implies Cond(P(A)) for

A

::;

K: if

7r

is a collapsing map then

rr(P(A))

{A

n

7r(A)

I

A E 7r(P(K))}. Thus Cond(P(K)) ~ 2'\

=

A+

for all

A::;

K.

Definition.

Let A

=

(A, ... ) with A transitive and let a ::; A

n

OR and

F:A<w ---> A.

(1)

<

a-Cond(A; F) iff for all P, P II- "IX. X E CF & X

n

a E a --->

rrx(A) E V.

(2) a-Cond(A; F) iff for all P, P II-"IX. X E CF & a ~ X ---> rrx(A) E

V.

a-Cond(A) iff there is an F such that a-Cond(A; F) and

<

a-Cond(A) there

is an F such that <a-Cond(A;F).

< K+-Cond(A) is equivalent to K-Cond(A). Lemmas 1.1 and 1.3 hold for

these notions with no change of argument. We have the following refinement

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Theorem 1.7.1.

(1)

K-Cond(P(>-))

implies 26

=

5+ for all 5 E

[K, >-1

.

(2)

K-Cond(P(K))

iff 2"

=

K+.

Proof. Clear from foregoing arguments. .,

Next, a few results concerning the effect of condensation on embeddings.

Lemma 1.10. Let M be a transitive model of

ZFC

,

A E M, A ~ K. If

M

F

Cond(A)

and j : M --+ N is elementary, then

A

E N.

Proof. Let M

F

Cond(A

;

F)

.

Then N

F

Cond(j(A);j(F)).

It is easy to

check that j" K is closed under

j(F).

Letting X

=

j" K, X

n j(A)

=

j"

A.

So

A

= 7rx"j(A)

EN

.

.,

The next two corollaries are immediate.

Corollary 1.11. If condensation holds in M and j : lvf --+ N 1S

elemen-tary, then M ~ N. .,

Remark: This is best possible: one cannot prove in

ZFC

that

M

=

N.

For example, assume V = L[J.Ll with K the unique measurable cardinal. We

will see shortly that condensation holds in V". Let X

-<

V" be Jonsson.

Thus

IXI

=

K and K \

X

-1= 0. The inverse of 7rx is a non-trivial elementary

embedding j : M --+ V" with critical point a

<

K. Condensation holds in

M.

Mis cofinal in

V".

And

M

-1=

V"

since a is not measurable. Stretch this to a class embedding using the embedding induced by J.L.

Corollary 1.12. If condensation holds then there are no measurable c

ardi-nals. .,

Corollary 1.13.

Cond

(w2)+2

w1

=

W2

_implies_{that ther}_e_are_no_precipitious

ideals on WI.

Proof. Let I be a countably complete ideal on WI and let j : V --+ M =

VW1IG

be the generic embedding. Suppose that M is well-founded. Then

W2 v

~

j(wl

V)

<

j(W2

V)

<

W3 v,

since

2W1

=

W2

.

In V

,

let

A

~

W2

code a

well-ordering of length

a

E (j

(W2

V),

W3

V). Since j

(W2

V) is a cardinal in

M

,

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§1.3 Two examples

The rest of this section is given to discussion of two examples: briefly,

condensation in L[Jl]' and in more detail, condensation in HODL(R) under suitable hypotheses.

Example 1. In L[Jl] the following hold:

(1) Cond(V,,+d (2) -,Cond(Jl).

L[Jl]

F

-,Cond(Jl) by lemma 1.10 (here regarding Jl

=

Jl

n

L[Jl] E L[Jl]).

Digression: one may take the statement Cond(Jl) to be Cond(A) for any

transitive structure A in L[Jl] such that L[Jl]

=

L[A]. To be specific, take the

simplest. Let A

=

Jl U Ii and A

=

(A, E). Then L[Jl]

F

-,Cond(A). Since AC holds in L[Jl] and L[Jl]

F

2"

=

Ii+, equivalently L[Jl]

F

-,Cond(I\:+, D) for any D ~ I\:+L[I') in L[Jl] which codes Jl. Although, in general there seems no reason to suppose that the failure of condensation for every set of ordinals

coding a set

S

implies the failure of condensation for

S

(or for

TC(S))

.

Thus

one might ask the following:

QUESTION. 13 it consistent that Cond(P(I\:)) holds and Cond(A) fails for

every set of ordinals coding P( 1\:), for some I\:?

Choice would have to fail in L(P(I\:)) in a model giving a positive answer.

For Cond(V,,+d, first argue that Cond(Va) holds in L[Jl] for 0'

<

Ii. This

is an easy application of techniques of Kunen

[K].

Let

B

>

I\: be large, and

Va E X -: Le[Jl], 0'

<

Ii. Let 7r =

trx :

X ----+ Li/[p], Li/[p] is clearly

iterable. So let 8

>

1\:+ be regular and iterate L[Jl] and Li/[p] up to L[F6] and

L-y[F6] respectively, where F6 is the club filter on 8. Since 0'

<

Ii, Li/[p]

F

\7r(Va)\

<

7r(Ii). So 7r(V,,) is fixed in the iteration of Li/[p], hence an element

of L-y[F6] ~ L[F6] ~ L[Jl]. This argument is independent of setting. In

particular it holds in generic extensions of L [Jl]. So L [Jl]

F

Va

<

Ii C ond( Va ).

Now let j : L[Jl] ----+ L[j(Jl)] be the ultrapower embedding. L[j(Jl)]

F

Va

<

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L[J.l]

F

Cond(V:~~I')J)

.

Finally,

Cond(V~~I)

=

Cond(V:~~I')J)

.

SO

L[J.lJ

F

Cond(V,,+1 ).

Applying theorem 1.7.1, the GCH holds for cardinals .A ~ /'i, in L[J.l]

.

The

usual argument that the GCH holds in

L[J.l]

for .A

2:

/'i, establishes that for .A 2: /'i" /'i,-Cond(P(.A)) holds. Note the duplication at /'i,.

Example 2. Let ('J) be the following statement: V

=

L(JR)

+

Scalep::i)+

There are no uncountable sequences of reals. With ZF

+

DC as background

theory, ('J) implies that HOD

F

Cond(HOD,,), where /'i,

=

WI and HOD" =

HOD

n

V".

Thus for the rest of this discussion tend to assume Z F

+

DC is in effect.

"Scale(L:i)" is the statement that every L:i relation on JR has a ~i-scale. It is a direct consequence of Scale(L:i) that every L:i set is the projection of a

tree in HOD. That is, if A ~ JR is L:i, then there is a tree T ~ (w x .A) <w for

some.A with T E HOD and A = p[T] = {x E Ww

I

~f

:

W --+ .A. "In (x

r

n,

f

r

n) E T}. (And something similar goes for L:i relations A ~ JRn. We identify

JR with ww .) In other words every L:i set is Suslin over HOD. (Assuming

V = L(JR), Scale(L:i) is equivalent to the statement that every L:i set is

Suslin over HOD.) For the needed background from descriptive set theory

see [M], [M-S], [S].

It might be worthwhile to put things somewhat into context before

pro-ceeding with the argument. Woodin has proved that 'J is equivalent to

ADL(JR). Assuming ADL(JR), HOD and

L[J.l]

are alike in that

condensa-tion holds up to the least measurable. HOD is the richer model, of course.

Woodin has shown that () is Woodin in HOD, where () is the supremum of

order-types of prewellorderings of JR. HODwl is an analogue of

LWI

corre-sponding to the pointclass L:i. The fact that HODwl is a model of

conden-sation supports the analogy.

Assume 'J and let /'i,

=

WI' Since there are no uncountable sequences of

reals, /'i, is inaccessible in HOD. HOD"

F

ZFC, hence can be identified

with its sets of ordinals. Also, if P E HOD" then there are filters G ~ P

(19)

that sets of reals claimed to be I:i below really are. This is straightforward

using reflection in the L(IR)-hierarchy and section 1 of [S], in particular that

(I:i)L(IR)

=

I:l

(L(IR),

{!R})- i.e., in

L(!R),

I:i sets of reals are those which are I:l definable with IR as a parameter.

Choose a reasonable coding of countable ordinals and their subsets by

reals: II~ sets W, C coding countable ordinals and their subsets respectively.

If

x

E

W

let

(J;c.

be the ordinal coded by

x

.

If

x

E

C

let

U;c.

be the set coded by

x.

Let

H

=

{x

E C

I

U;c.

E

HOD}. H

is a I:i set of reals which

codes

HOD

n

[K]<"

(essentially

HOD,,).

Let

<OD

be the well-ordering

of ordinal definable sets given by:

U <OD

V iff for every

(cx,(J,¢)

with

V

=

D

(cx,(3,¢)

there is

(cx

'

,(3

'

,¢

'

) <

l

ex

(cx

,(3,¢)

with

U

=

D

(cx

'

,(3

'

,¢

'

)

where

D

(cx,(3,¢)

=

{

x

I

L

o(

!R

) F

¢(x,(3)}.

(Formulas ¢ are identified with

elements of w in a suitable way.)

<O

D

induces a I:i prewellordering on

H

by x

:S

y iff x, y E

H

,

and

U

x

:SOD

U

y . Also, there is a IIi relation

:S

*

such that

y

E

H

implies that

{

x

I

x

:S

*

y}

=

{

x

I

x

:S

y}

C;

H.

Thus

:S,

in turn, yields

a

I:i-nor

m

on H, ¢ : H ---+

A,

for some

A.

(See

[M].

)

Define A C; !R2 _by

(x,

z)

E A iff

z

codes an initial segment

ofP«(3x)nHOD

under

:SO

D

.

A is

I:i-To see this let

(z )n(

k) =

z(

(-,

.

)), (-,

.

)

a recursive pairing function.

(x, z)

E A iff

Vn [U(Z)n

E

H

&

Vy(U

y C;

(3x

&

y

:S

*

(z)

n

---+

3mU

y =

U(Z)m)]

'

By

Scale(I:i) there is a tree

T

C; (w2 _X

A

)<w

_for_so_m_e

A

_with_A

=

p[T].

Let 8

>

K

be large and let

K,

T

E X ~

HOD

6 , 7r : X ---+ M,

7r(K,

A,

T

) =

(~,~,

T

)

and M =

7r(HOD,,)

.

M is a transitive model of ZFC, hence can

be identified with its sets of ordinals. Under this identification:

Claim. M is an initial segment of U{J<;c

P

«(3)

n

HOD

under

<OD

·

Let

P

{J

= Coll(w, (3) and If-{J be the associated forcing relation. We have

HOD

F

V(3

<

K,U

C;

(3

.3

1'

If--y

(

3x

,z,n.

(3

=

(3x

&

(x,z)

E

p[T]

&

U

=

U(Z)n)'

To see this let l' =

(2{J)

HOD

and G be P-y=generic over

HOD

.

In

HOD[G]

pick

x

coding (3 and

z

coding

P«(3)

n

HOD.

Then

(x, z)

E

p[T]

.

Hence by absoluteness,

HOD[G]

F

(x,

z)

E

p[T].

(T(x,

z)

is well-founded iff well-founded in

HOD[G]

.

And

(x,

z)

E

p[T]

iff

T(x

, z)

is not well-founded.

(20)

f3.

3{11-,. (3x,z,n.

f3

=

f3r

& (x,z) E p[T] & U

=

U(Z)n). Let U E M,

U ~

f3

<

K. Pick a suitable {

<

K and let G be P,.-generic over M. In

M[G] pick (x, z) E p[T] with U = U(Z)n for some n. Let

f:

w --+

>.

witness

(x, z) E p[T]. Define f : W --+ A by fen)

=

7r-1Cf(n)). Applying 7r-1

pointwise for all n, (x

r

n,z

r

n,f

r

n) E T. Thus (x,z) E p[T]

=

A. So

(z)n E Hand U

=

U(Z)n E HOD. At this point we have M ~ HOD. Now

let V ~

f3

'

<

K, V E HOD, V <OD U. Clearly one can take

f3

>

f3

'

above. Thus for some m, U(Z)m = V. So V E M[G]. But this does not depend on

the particular generic G chosen. Thus V E M and the claim is established.

It is immediate from the claim that ME HOD. That HOD

F

Cond(HOD,,)

(21)

Section 2

In this section we present the main result of this thesis, theorem 2.2, that

condensation implies O. The argument supplies a new proof of lemma 1.6.

Also, we discuss the relationship between condensation and 0*, and present

a gap-2 version of 0* which implies Cond(w2)'

If I\. is a regular cardinal and E

<;

I\. is stationary, recall that 01< (E) asserts

that there is a sequence (Sa 1 a

<

1\.) such that (1) for all a, Sol

<;

a and

(2) for every A

<;

1\.,

{a

EEl A

n

a

= Sol} is stationary in 1\.. 01< = 0,,(1\.).

If I\.

= WI the subscript is

supressed. O,,(E) is equivalent to the assertion

that there is a sequence (Sa 1 a

<

K)

such that (1) for all a, Sol

<;

pea) and

ISal

:S

a and (2) for every A

<;

1\., {a EEl A

n

a E Sol} is stationary in K

[D].

Let A

=

(A, E, ... ) be a structure with A transitive. For A, F E X let

7rx : X --+ Mx be the transitive collapse of X, Ax = 7rx(A), Fx = 7rx(F).

The next lemma is just a special case of lemma 1.1.

Lemma 2.1. A33ume Cond(A; F). Let 8 be 3uch that Ve

F

Cond(A; F)+T with T a 3ufficiently 3trong fragment of ZFC. If

A

,

F E X

<;

Y and X, Y

-<

Ve then Ax E My.

Proof. Let X = 7rY"X. My

F

Cond(Ay;Fy)

+

T and X

n

Ay is closed under Fy. Applying lemma 1.1 (and the remark following its proof) with

M = My we must have Ax = _{7r x (}Ay) E My. .,

Theorem 2.2. Let <l be a well-ordering of HW2 of minimal order-type and

A

=

(Hw21 <l). A33ume Cond(A). Then for every 3tationary E

<;

WI, O(E)

hold3.

Proof. By theorem 1.7, 2w,

=

W2' SO IHw21

=

W2 and <l has order-type W2'

Every initial segment of <l is an element of H_W2'

Let F : H::'w --+ HW2 witness Cond(A) and take Ve as in lemma 2.1, with

cf(8)

>

WI. Let

H

: Ve<w

--+

Ve

be a skolem function for

(Ve,

{A,F}). So

(22)

clH(X)

=

H"X<w.

Let C =

{X

E

[Vu]W

I

X -( (Vu,H)}.

C is club in

[Vu]w.

If

X

E C then

X

E

CH, X

n

HW2

E

CF

n

[HW2 ]W

and

X

n WI is an ordinal

ax

<

WI·

For

X -(

Vulet

X+

= {J(ax)

I

j E W1VnX}. If

X

E C then

X+

=

clH(X

U {ax}).

X+

~

clH(X

u

{ax}) is clear. For the reverse inclusion

let

P

E

[Vu]<w

and define jp(a)

=

clH(pU

{a}) for a

<

WI. jp E

Vu

since

cj(B)

>

WI. If P EX E C then jp E

X

.

Now let u E

clH(X

U {ax}) =

H"(X

U {ax}

)<w

.

Then u

=

H(s)

for some

s

E

(p

U {ax}

)<w

with p EX. Thus u E jp( ax) E

X+

.

Furthermore, each jp( a) is countable. So there is

agE

X

such that for all a

<

WI, g(a) : W ~ jp(a). Then g(ax) E

X+

.

Also W ~

X

~

X+

.

Thus jp(ax) ~

X+

and u E

X+.

It follows that if

X

E C then A,F E

X+ -( Vu.

Hence by lemma 2.1,

for X E C, Ax E Mx+. Thus Ax = 7rx+(J(ax)) for some j E X . Let

a

=

ax, Ax

=

(Ax,<lx). By elementarity j(a)

=

(AO',<lo) where Ao

F=

a

=

WI

+

"J am H

W2 "

.

<lo well-orders Ao and every initial segment of

<lo is an element of Ao. Also a is countable in

X+

.

So 7rx+ fixes every element of Ao n

X+.

So Ao n

X+

is a <lo-initial segment of Ao and Ax

=

7r x+(J(a)) = j(a)

I

X+

= (Ao n

X+,

<lo

I

ax+) where <lo

I

f3

is the length

f3

initial segment of <lo. (Note: possibly Ao is countable, in which case

7rx+(J(a))

=

j(a)

=

Ax. We cannot, at tl i point, prove that Ao is

countable, however. It is clear that

e(

<lo)

:S

WI. Whether

e(

<lo)

<

WI

relates to the question of whether condensation implies 0·. See theorem

2.4).

Let E ~ WI be stationary. The set

So

=

{X

E C

I

ax E E} is stationary

in

[Vu]w.

By Fodor's theorem there is a single j and a stationary

S

~

So

such that for

XES,

Ax = 7r;u(J(ax )). Define a O(E) sequence using this

j in the usual way: given 5

r

a let (50' Co) be the <lo-least pair of subsets

of a such that Co is club in a and

f3

E CO' n E implies that 50 n

f3

-j. 5{3,

if such exists, and 0 otherwise. Let 5 = (50

I

a

<

WI). Suppose that 5

fails to be a O(E) sequence. Let (U, D) be the <l-least counterexample. Take

(23)

that S

r

a is not O(E n a)," where a = ax. Since XES, <lx is an initial segment of <lao Thus (U

n

a, Dna) satisfies the definition of

(S"t>

Co). So

Una

=

So. On the other hand, since DE XES, a E D n E, contradicting

the choice of U and D. So S is a O(E) sequence. .,

Remark: One can avoid the use of X+ above (and hence the need to

assume that cf(8)

>

wd

as follows. Let X· = clH(X U {ax}). Then

X <;;:

x·

and A,F E

x·

-<

Vo. By lemma 2.1, Ax

=

7rx.(H(s» for some

S E (X U {ax})<"'. For some p EX, s E (pU {ax})<"'. Thus, although

s

¢

X, there is a parameter matix m E X such that s

=

m( ax). Fix this m

on a stationary subset of So, and define

f

on WI by

f(a)

=

H(m(a»

.

Then

f

can be used as above to define a O(E) sequence.

Theorem 202(b). Let" be a regular cardinal and <l be a well-ordering of

H,,+ of minimal order-type. Let A = (H,,+, <l) and a3.mme

<

,,-Cond(A)

hold3. Then for every 3tationary E <;;: ", O,,(E) hold3.

Proof. The proof, with the obvious changes, is identical. In choosing Cone

must stipulate that X

n

"

is an ordinal. .,

Corollary 2.3. A33ume Cond(P(A». Then for every regular K

<

A and

3tationary E <;;: ", O,,(E) hold3. .,

The proof of 2.2 contains a proof that Cond(H"" , <l)

+

2"'1

=

W2 implies

CH. This by itself is no improvement on theorem 1.7 which assumes only

Cond(P(w». But the argument is of a different type. In fact the method of argument supplies another proof of lemma 1.6 (hence, that Cond(P(w» is

sufficient for C

H).

Proof of lemma 1.6 (3rd). Let a

<

WI, A <;;: P(a) and assume Cond(A).

The only new observation needed is that if R is a relation on A, then also

Cond(R) holds. (This is not necessarily true if a 2 WI. E.g., if there is an

inner model with a measurable cardinal then the measure in

L[fll

gives a

counterexample. The large cardinal assumption is probably irrelevant.) To see this let R <;;: An and choose F witnessing Cond(A) so that if X E CF

(24)

Pea) we have 7rx(R)

=

{a I a ERn X}

=

{a I a ERn (An n X)}

=

{a I

a ERn (A n x)n}. Thus 7rx(R) is definable from R and A n X

=

7rx(A).

So Cond(R) holds.

Now assume IAI ~ W2. Let <l be a well-ordering of a subset of A of length

W2' By the preceding Cond( <l) holds. As in the proof of 2.2 find C C;

[V8]W

such that X E C implies that <l,F E X+ -( V8 so that <lx= 7rx(<l) E Mx+.

So there is an f E X such that <lx= 7rx+(f(ax» = f(ax)

n

X+. Since

Mx

F

ot( <lx) = W2 and ax is countable in _{M x +, Mx+}

F

ot( <lx) ::; WI' Thus by elementari ty f (a x)

=

<l0' x orders a subset of P( a) and ot(f (a x» ::;

WI. Now fix

f

on a stationary set

S.

W.l.o.g., for all a ::; WI, If(a)1 ::; WI.

Let a E fldC<l). Take XES with a E X. Then a E fld(<lO'x). So

f ld ( <l) C; U {J ld ( <l0') I a

<

wd·

This implies that I <l I

=

WI contradicting

the choice of <l. Thus IAI ::; WI' ..,

Let K, be a regular cardinal, F a normal filter on K,. E C; K, is F-positive

if K. \ E ~ F. O*(F) is the statement: there is a sequence (So

I

a

<

K.)

such that (1) for all a

<

K,,50' C; pea) and 150'1::; a, and (2) for all A C; K"

{a

I

A

n

a

E

Sa}

E F. O~

=

O*(Cub,,) where Cub" is the club filter on K,.

wCCCK.) is the statement that for every A

=

CA, ... ) with K.+ C; A there is an a such that sup{otCX

n

K,+)

I

X -( A & X

n

K, =

a}

~ a+. wCC(K.) is

the weak Chang conjecture for K,. wCC is WCC(WI)'

It is easy to see that for every normal filter

F

on K, and F-positive

E

,

O~ ==} O*(F) ==} O,,(E) ==} 0". Here are some facts relating O~ and

wCC(K.). (See [B], [D-KJ, [D-L].)

(1) wCC(K,) implies ...,O~.

(2) Assuming V

=

L, wCC( K.) iff ...,O~ iff K, is ineffable.

(3) If K, ~ W2 is a successor cardinal and wCC( K,) holds, then

at

exists.

As a corollary to the proof of 2.2 we have:

Theorem 2.4. Let A be as in 2.2. Assume Cond(A) and ...,wCC. Then

(25)

Proof. Take

f

as In the proof of 2.2. So on a stationary set S ~ So =

{X

I

ax E E}, Ax = 7rx+(f(ax)). Let £ = _{{p(C A}nS)

I

A ~ wd where C A

=

{X

I

A E X} and p(U)

=

{ax

I

X E U}. Then £ generates a non-trivial normal filter

F

on WI with

E

F.

p(CA

n

S) ~ peS) ~ peSo)

=

E. So E E:F. To check that F is normal let Ba E F for a

<

WI and

B

=

6aBa

=

{.B

I

.B

E na<f3 Ba}· Let p(CAa

n

S) ~ Ba. And let A

code (Aa

I

a

<

WI) so that on a club C ~ (VB]"', A E X E C implies that (Aa

I

a

<

WI) E X. If X E CA

n

S then for a

<

ax, Aa E X so that

ax E _p(CAa

n

S) E Ba. Thus ax E B and p(CA

n

S) ~ B. This gives

B

E :F.

F

is clearly nontrivial.

Since wCC fails, by shrinking S if necessary one can, for each a

<

WI,

bound the order-type of X

n

W2 for XES with ax = a independently of

X . Let b (a) give this bound. One can assume also that b E X so that

b(ax)

<

ax+. So for XES, Ax

=

7rx+(f(ax))

=

I(ax). I can be

chosen so that

I

(a) is countable for all a. It is now easy to check that

Sa

=

pea)

n

I(a) defines a O*(F) sequence. .,

Again nothing in this argument depends on the fact that WI is the least

uncountable cardinal. Thus more generally:

Theorem 2.4(b). Let

A

=

(H", <J) where", i3 regular and <J i3 a well-ordering of H" of minimal type. A33ume

< ",-Cond(A)

and .wCC(",). Then

for every 3tationary E ~ K. there i3 a normal filter F 3uch that E E F and

O*(F) hold3.

This is a partial converse of (1). By (2) the full converse holds in L.

Theorem 2.4 itself is a little odd. wCC( "') is a large cardinal property in L.

By (3), in K if wCC( "') holds then'" is at least inaccessible. And 0* holds

(hence wC C fails) in the model of exam pie 2 in section 1. So it seems likely

that condensation simply refutes wCC and wCC(",+) more generally.

0* has the following equivalent formulation: there is a sequence S

=

(Sa

I

a

<

wt) such that (1)

ISol

= wand (2) for every A ~ WI, {X E

[H"'2]'"

I

7rx(A) E Sax} is club. Boost this a bit to get a "gap-2" strong diamond

(26)

Definition. 0 2 _is_the_statement:_{there is}_{a sequence}

5

₌

(50/

I

_a

<

WI) such

that (1)

150/1

= wand (2) for every

A

<:::; W2, {X E [HW3]W

I

7rx(A)

E SO/x} is

club.

Theorem 2.5.

(a)

L, L[J.L]

F02.

(b) 02 _implies_Cond(w2)'

Proof.

(a) In L, let

50/

=

L{3 where

/3

is least such that LfJ

F

a

is countable.

It is easy to verify that this defines a 02 _sequence._{So turn}_to_L[J.L].

Working in L[J.L]' let K be the measurable. Let F

=

Cubwi ' Note that

IR <:::; L[F]. So WI L[F)

=

WI' For a

<

WI let

7]0/

be the least T/ such that

L'I[F]

F lal

= W,

/0/

= WI L'a [F) and let

/30/

be the least

/3

such that L{3[F]

F

I/O/I

=

w. Take 50/

=

(HC)Lpa[F). This will do.

We show that

50/

is countable. Let a <:::; W code T ~

a.

Let ()

>

K be large

and a E X ~ Le[J.L]' with

IXI

=

w. 7rX : X ---+ Lo[p] with K

=

7rX(K). Thus

a

<

WI L/[il)

<

K

<

WI and Lo[p] is iterable. Iterate Lo[p] up to Lo[F]. Then

{j ~ T/O/ and /0 :::; WI L6[F)

=

WI L,;-[il). So

/0/

is countable and we may assume

that T ~

/0/

'

So {j ~

/30

'

Now

50/

=

(HC)LPa[F) <:::; (HC)L6[F)

=

(HC)Li[il) is countable.

Let A <:::; W2 and C

=

{Y

n

HW3

I

A E Y ~ Le[J.L]}. C is club in [HW3]w, Let X E C and X = Y

n H

_W3'

Ax

=

7rx(A)

=

7ry(A).

Let

a

=

ax

=

ay.

7ry : Y ---+ Lo[p]

F

a

= WI. Iterate this model to Lo[F]. Also Lo[F]

Fa

=

WI' Let v = W2 L6 [F) = W2 L/[il). Then

Ax <:::;

v:::; WIL.a[F) = /0' Since v

<

K,

v and

Ax

are fixed in the iteration of Lo[p]. So

Ax

E Lo[F] <:::; L{3a [F]

F

Ivl

=

w. It follows that

Ax E

50/'

So

(50/

I

a

<

WI) is 0 2•

(b) Let A <:::; W2 and assume C ond( A) fails. A review of the proof of lemma

(27)

Section 3

This section addresses the problem of obtaining Cond(A) by forcing. One

would like, assuming GCH, to add Cond(A) with set forcing, without

col-lapsing cardinals. The immediate motivation for this is corollary 1.13. A small forcing notion which adds Cond(w2)

+

2w

, = W2 shows that the

ex-istence of a precipitous ideal is not a consequence of large cardinal axioms. This problem has been attacked from more than one direction now. And to our knowledge is still open. Thus we speculate that the difficulty of this problem is related to the difficulty of forcing Cond(A). But the matter of

forcing Cond( A) retains its interest in spite of this question about precipitous ideals. It was raised in a slightly different guise in Lee Stanley's thesis [Stn].

Stanley shows how to force the existence of higher-gap morasses, noting that these forced morasses lack some of the properties of the "natural" morasses which can be constructed in L, f{ and higher core models. (See [W].) These properties are precisely the ones needed to verify that the forced morasses have condensation. This is notable since a gap-2 morass at WI which has

condensation kills all precipitous ideals on WI.

Jensen's coding theorem shows that condensation can be added globally without collapsing cardinals, assuming eCHo The argument is essentially

top down or "Easton style" using a class partial order to produce a model

of V

=

L[a], a

<;

W. The basic strategy is to code A

<;

1\:+ by B

<;

1\:, in the

sense that A E L[B]. The distributivity arguments involved in adding this B require the assumption

V

=

L[A],

hence the success of coding from 00 down

to 1\:+. Some such assumption is evidently necessary. For example, if

>.

is

measurable, I\:

<

>.

and

A

<;

1\:+ is such that (I\:+)L[AJ

=

1\:+, then there is no P E VA which adds B

<;

I\: with A E L[B] without collapsing 1\:+. The reason

is that BI exists after forcing with P. So (I\:+)v = (I\:+)L[AJ S; (1\:+)L[8J

<

(1\:+) V[GJ. Thus one cannot in general apply the preceding strategy locally to add Cond(A). On the other hand if there is an object C with condensation

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§3.1 Coding to add Cond(A).

Recall almost disjoint (adj) coding. Let b

=

(b"

I

v

<

11:+) be an almost

disjoint sequence of subsets of 11:: if v

t

J-l then

Ib"

n

bl'l

<

11:. Let P consist

of pairs (b, u) with b ~ 11:, U ~ 11:+ and

Ibl,

lui

<

11:. And let (b, u) SA

(b,

u) if

b ~e b, u ~ u and for v E unA, (b\b)nbv

=

0. Let PA

=

P(b, A)

=

(P, SA).

PA is II:-closed and, if 11:<1<

=

11:, satisfies the II:+-c.c. PA adds a set B ~ II:

such that v E A iff

IB

n

b"

I

<

11:. If this last condition is fulfilled, say that the pair (b, B) codes A.

Lemma 3.1. If (b, B ) ha3 conden3ation and code3 A, then A ha3

conden-Mtion.

Proof. A E L[b, B]. -l

Lemma 3.2. (GCH) Let bl, ... ,bn be adj 3equences with conden3ation,

bk

=

(b~

I

v

<

Wk+I), bt ~ Wk. Let A ~ Wn+l. Taking Bn+1

=

A, supp03e

that Bk i3 P(b k , Bk+d-generic over V[Bn, ... , Bk+d, 1

<

II:

S

n. then A

ha3 conden3ation in V[Bn, . .. , Bd.

Proof. By induction working in V[B n, ... , Bd. BI ~ WI has condensation.

Assume B k has condensation. (b k, B k) codes B k+ I. SO B k+ I has

condensa-tion by 3.1. -l

Lemma 3.3. (CCH) A33ume that W2, ... ,Wn+1 are accessible in LICl for

30me

C

such that Cond(C) hold3. Then for A ~ Wn+1 there i3 a p.o. P

such that if G is P-generic over V, then V[G]

F

Cond(A) and P preJerveJ

cardinalJ.

Proof. Let Wi+1

=

(tSt)L[Cl, 1

SiS

n. Choose a ~ Wn such that for

1

SiS

n, _{ItS i}IL[anw;J

=

Wi and let ai

=

a

n

Wi. L[C, ail correctly computes Wi+l. So there is an adj sequence bi E L[ C,

ad

for coding subsets of Wi+1 by

subsets of Wi. _{Note Cond(b l )}holds. At this point one could iterate lemma

3.2 getting Cond(ai) hence _{Cond(b i )}successively. But this is inefficient.

Instead just code from Wn+1 to WI using the bi'S. Code A by Bn ~ Wn using

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C has condensation and aI, BI

<;;:

WI, (C, aI, Bd has condensation. Thus

Cond(A) holds. The required P is the finite iteration which accomplishes

this. ...,

Lemma 3.3 can be improved upon considerably in two ways. The coding

can be pushed past WW ' And the condition "/\:+ is accessible in L[C]" can be

replaced by the condition: for some B,

The next two lemmas give examples.

Lemma 3.4. (GCH) A33ume Cond(C) and that for n

<

w, Wn i3

acce33t-ble in L[C]. Then for any A

<;;:

Ww , there i3 a partial order P which add3 Cond(A) and pre3erve3 cardinal3.

Proof. One can assume that L[A

n

_{wn] correctly computes Wn} and that

L[A

n

wn]

F=

16

n

l

=

_{Wn , where Wn+l}

=

(6;!")L[CI. Let An

=

An Wn. So

L[C, An] correctly computes Wn+I' And there is an adj sequence b n

=

(b~

I

v

< Wn+l)

E L[C, An] for coding subsets of Wn+1 by subsets of W_n. Let b n

be the <L[c,An]"least such sequence.

Define P as follows. A condition is P = (Pn

I

1

:s

n

<

w), pn

=

(b n , un) with bn

<;;:

Wn , Un

<;;:

Wn+l and

Ibnl,lunl

<

wn . p:S

P

iff for all n, Pn :S(bn+1,A n+1) Pn- i.e., for all n,

(1)

b

n

<;;:e

bn,u n <;;:u n

(2) If v E Un

n

(b

n+I ,An+1 ) then bn \

b

n

b~

=

0

where (5,

T)

= {21] 11] E 5} U {21]

+

1 11] E

T}. P

is as required.

The following sets are dense. For v

<

Wn+b E~

=

{p

I

v E un}. For

1]

<

Wn,v

< wn+l,D~'v

=

{p

I

supbn

>

1] and if v

rt

(bn+I,An+d then bn

n

b~ \ 1]

#

0}.

If p,

P

agree in their first components they are compatible. Thus E~ IS

dense.

Let PEP, 1]

<

Wn , V

<

Wn+l. Let Pk

=

Pk if k

#

n and let Un

=

Un. If V

rt

(b

n+l , A n+I ), let T = v. Otherwise let T E Wn+l \

(b

n+1 , A n+1 ) be

(30)

ij

=

SUp{T)v

I

v E Un

n

(bn+I,An+I)}' Since

Iunl

<

wn,ij

<

Wn. Now take

~ E b~, ~

>

T), ij and let bn =

b

n U

{o.

Then p

:s

p

and p E D~'v.

Let G be P-generic, Bn

=

U{(Pn)o

I

pEG}. Using the density of E~

and D~'v it is easy to check that for all n, (b_{n , Bn)}codes (B n+b A n+I). If

v E (Bn+I' A n+l ) pick pEG with v E (b n+I , A n+l ) and pi

:s

p,pl E G

n

E~.

If p:S pi then (bn,u n ) ::;(b~+l,An+l) (b~,u~). So since v E Un

n

(b'n+I,A n+I ),

bnnb~ ~ supb~. Thus Bnnb~ ~ b~ and IBnnb~1

<

wn. If v ~ (Bn+I' A n+I ),

let T)

<

Wn and take pEG

n

D~'v. Then bn

n

b~ \ T)

-I

0. Thus Bn

n

b~ is unbounded in Wn .

A E L[C,AI,Bd. To see this, working in _{L[C,AI,B I]}recursively decode

the sequence ((Bn, An)

11

:s

n

<

w) using the fact that bn is

<

L[c,Antleast,

and that (bn,Bn) codes _{(Bn+I,A n+I ).} So the sequence is in L[C,AI,Bd.

Then obviously T) E A iff 2TJ

+

1 E (Bn' An) whenever TJ

<

_Wn. As before,

since C has condensation and AI,B I ~ WI, (C,AI,B I ) has condensation.

Since A E L[C, AI, Bd, Cond(A) holds.

It remains to show that

P

preserves cardinals. Let

P

n

=

{p

r

[n,w)

I

p E

P} and :Sn=:S

r

Pn. For D ~ _Wn,let Pfj

=

{p

r

nip E P} and define

:So on Pfj by p :So p iff (1) for 1

:s

i

<

n - I, (bi , Ui) :S(i;,+l,A,+l)

(b

i , Ui)

and _{(2) (bn-I,un-d :S(D,A n) (bn-I,un-d.} Now let p(n) consist of those

pEP with SUp Un_1

<

_{SUp (b n,}0). p(n) is dense in P. To see this, let

pEP. c

=

Wn \ U{b~

I

v E un} is unbounded in Wn since

Iunl

<

Wn and (b~

I

v

<

Wn+l) is adj. Let 8 be least in c \ SUp Un-I, b~ = bn U {8},

p~ = Pk if k

-I

n, and p~ = (b'n' un). Then pi E p(n). For v E Un,

b~ \ bn

n

b~ = {8}

n

b~ = 0. So p~ _{:S(b n+}1,An+l) Pn. Since pi differs from p

only in the n-th coordinate, pi

:s

p. So p(n) is dense. Claim: for p,p E p(n),

(1) p

r

[n,w) II-Pn p

r

n E PIL, and

(2) p:S P iff p

r

[n,w) :Sn P

r

[n,w) and p

r

[n,w) II-Pn P

r

n :SEn p

r

n.

(1)

is trivial. For

(2),

let p

:s

p. p

r

[n,w) :Sn P

r

[n,w) is clear. p

[n, w) II-P_nP

r

n :SEn p

r

n will hold if for all q :Sn p

r

[n, w), letting

b~

=

(qn)O, Pn-I :S(b~,An) Pn-I' Note that if p E p(n) and B

2.

bn, then

Un-I

n

(B,An)

=

Un-I

n

(bn,An). Thus for any pEP, Pn-I _{:S(B,A n)}

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pn-I :S(b~,An) Pn-I· For the converse, p

r

[n,w) If-Pn P

r

n :San P

r

n implies that pn-I :S(bnoAn) Pn-I, hence that pn-I :S(i;noAn) Pn-I. Adding the

condition p

r

[n,w) :Sn P

r

[n,w) will ensure that p:S

p.

The claim follows. So for every n, forcing with P is equivalent to forcing with the two-step

iteration Pn

*

PIt. Pn is wn-closed. And since conditions in PJj are

com-patible if their first components are, Pn If- PIt has the wn-c.c. So Wn is not

collapsed. Since for all n

<

W, Wn is not collapsed, neither is WW. And P has the Ww+I-C.C. So cardinals> Ww are preserved. -1

Lemma 3.5.

(GCH)

Assume Cond(C). Let"

=

W2 and assume that for

some B ~

"+,

(*)~,B holds. Then for any A ~ W3 there is a P which adds

Cond(A) and preserves cardinals.

Proof. One can assume that A E L[C, B] and that H"

=

L,,[B

n

"-1-

The first step is to add D ~ K. such that B E L[C, D]. Then by lemma 3.6 below

one can add Cond(D). In the resulting model Cond(A) will hold.

Add D exactly as in [B-J-W] (p.9ff). In L[C,B

n

K.]let h : H" --+ K.

be a bijection. For any b ~ K. let S(b) = {h(b

n

8)

I

8

<

K.}. If b

=I

b',

then S(b)

n

S(b' ) is bounded in K.. Define a sequence (b.

I

8

<

K.+), b. ~ K., as follows. Given (b"

I

lJ

<

8), let b. be the <L[c,Bnwleast b ~ K. such

that b

=I

b" for lJ

<

8. b₆exists since b

r

8 is uniformly definable from C

and B

n

8

and L[C, B

n

8]

F

181

= K.. Now let D code B relative to the sequence (S(b.)

I

8

<

K.). Then B E L[C, B

n K.,

D]. To see this, working in L[C, B

n

K., DJ, B

n

K. is available. Given B

n

8, 8 E B iff D

n

S(b.) is bounded. And

b.

is uniformly defined from

B

n

8 and C. So

B

n

8

+

1 is

decided. Since the definitions are uniform, the sequence (B

n

8

I

8

<

K.+) is

in L[C, B

n

K., D]. Thus B E L[C, B

n

K., D]. Now code B

n

K. into D to get the required subset of K.. -1

The last two arguments are just direct adaptations from [B-J -W]. The

point here is that under certain conditions these arguments can be localized

to add condensation. Here is a guess at a more general fact of this sort:

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any A ~ ,\ there 13 a partial order P which add3 Cond(A) and pre3erve3 cardinal3.

"Proof". It is the need to ensure (* )~,A in order to code from

,,+

to " which

is responsible for introducing the assumption V

=

L[A] in [B-J- W].

We haven't seen any reason to develop this argument. Our immediate goal has been to force Cond(A) for some A ~ W3 which computes W3 correctly

thereby killing all precipitous ideals on WI. SO we are more interested in the

question: when does C exist? Or, are there large cardinal assumptions which

imply that the conditions of lemma 3.5 cannot be satisfied?

§3.2 Condensation and morasses.

Given some condensation one can add more. For A ~ W2 condensation

can be added outright. An (WI, I)-morass has condensation. Only "coarse" properties of the morass are needed for this. The partial order used in the

following argument is distilled from the one Jensen used to add an (WI,

1)-morass [Stn]. It adds what we will call an (WI, 1, A)-weak-morass, for the

purpose of discussion later.

Lemma 3.6. (CH) For any A ~ W2 there i3 a partial order P 3uch that if

G i3 P-generic then V[G]

F

Cond(A), and P pre3erve3 cardinal3.

Proof. Fix A ~ W2. Let F : H:5:,w --+ Hw, be a skolem function for Hw,

and CF

=

{X

n

W2

IIXI

=

wand

X

is closed under

F}

.

For a

<

WI, let

C'F =

{X

E CF

I

ax = a} where, as before, ax =

xnwi.

Let Ax = 7rx"A, QA,F = Q = {Ax

I

X E CF}, and Qa = {Ax

I

X E C'F}. For a

<

WI and

II E (WI,W2), let X~ = clF(a U {II}), D" = {a

I

X~

n

WI = a}, 7r~ = 7rx~,

and A~

=

7r~" A. Note that ~ and ~e agree on C'F, and that (C'F,~) is a tree. Also, for each a, (Qa, ~e) is a tree with height::; WI.

Define P as follows. Conditions are pairs p

=

(s,

U) satisfying:

(1) s : d --+

Q

where d is a closed and bounded subset of WI, and for

(33)

(2) U

~ U{D" x {v}

I

v E

(WI,W2)}.

(3)

181

=

lUI

=

w.

(4) If (a, v) E U then A~ E sea).

(5) If (a,v) E U then (ap,v) E U. And if a ~ fJ ~ a p and fJ E dn D"

then (fJ,v) E U.

(s, U) ~ (t, V) ift ~ s and V ~ U. Any two conditions (8, U) and (s, V) are compatible. So P h

An Abstract Condensation Property

An Abs

ACKNOWLEDGMENTS

CONTENTS

Acknowledgments

IAI<w

IPII-VX.X

W -( VB, IWI

IWI vI

Definition.

Theorem 1.7.1.

K-Cond(P(>-))

Corollary 1.13.

Cond(V:~~I')J)

I:i-nor

= WI the subscript is

clH(pU

< ",-Cond(A)

Proof.

Section 3

Ibnl,lunl

Lemma 3.5.